On the dimension theory of metrizable spaces with periodic homeomorphisms

For every metric space $X$ with homeomorphism $a\colon X\to X$ of prime period $p$ ($a^p=e_X$) we construct a zero-dimensional metric space $P$ ($\dim P=0$) with homeomorphism $b\colon P\to P$ of the same period $p$, together with a closed mapping $f\colon P\to X$ onto $X$, commuting with $a$ and $b$, such that $\operatorname{Ord}f\leqslant \dim X+1$ if $X$ is finite-dimensional and $\operatorname{Ord}f<\infty$ if $X$ is countable-dimensional.
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